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NASA Technical Memorandum 103830 



Tables of Square-Law Signal 
Detection Statistics for Hann 
Spectra with 50% Overlap 



Stanley R. Deans and D. Kent Cullers 



(NASA-TM-103830) TABLES OF S?^UARe-LAW N92-11745 

SIGNAL OETECTION STATISTICS FOR HANN SPECTRA 
WITH 50 PERCENT OVERLAP (NASA) 20 p 

CSCL 12A Unclas 
G3/65 0048768 



September 1 991 



fWNSA 

National Aeronautics and 
Space Administration 



NASA Technical Memorandum 103830 



Tables of Square-Law Signal 
Detection Statistics for Hann 
Spectra with 50% Overlap 

Stanley R. Deans, University of South Florida, Tampa, Florida 
D. Kent Cullers, Ames Research Center, Moffett Field, California 



September 1991 



rVI/NSA 

National Aeronautics and 
Space Administration 

Ames Research Center 

Moffett Field, California 94035-1000 



SUMMARY 

The Search for Extraterrestrial Intelligence, currently being planned by NASA, will require that 
an enormous amount of data be analyzed in real time by special-purpose hardware, and it is expected 
that overlapped Hann data windows will play an important role in this analysis. In order to understand 
the statistical implications of this approach, it has been necessary to compute detection statistics for 
overlapped Hann spectra. Tables of signal detection statistics are given for false alarm rates from 10" ''^ 
to 10"^ and signal detection probabilities from 0.50 to 0.99; the number of computed spectra ranges 
from 4 to 2000. 

INTRODUCTION 

The Search for Extraterrestrial Intelligence (SETI), currently being planned by NASA, will require 
that an enormous amount of data be analyzed in real time by special-purpose hardware, and it is 
expected that overlapped Hann (sometimes called Hanning) data windows will play an important role 
in this analysis. (A discussion of signal processing in SETI, and arguments for using overiapped Hann 
spectra, are given in refs. 1 and 2, and a good technical discussion of overiapped Hann spectra is given 
by Harris in ref. 3.) In order to understand the statistical implications of this approach, it has been 
necessary to compute detection statistics for overlapped Hann spectra. The main purpose of this report 
is to present tables of signal detection statistics for overlapped Hann spectra and demonstrate how to 
use them. Tables are given for false alarm rates from 10"^"^ to 10"' and signal detection probabilities 
from 0.50 to 0.99; the number of computed spectra ranges from 4 to 2000 (tables 1-14). 

This work was funded in part by NASA #NCC 2-580 at Ames Research Center. 

METHOD OF COMPUTATION 

The statistical parameters are found by integrating the characteristic function to find the cu- 
mulative distribution function F. Given the characteristic function, there are several ways to find the 
cumulative distribution function and the probability density function / (ref. 4, chap. 29). Three different 
methods have been used, as a cross-check to guard against programming mistakes. One of the more 
useful approaches is to compute the cumulative distribution directly from the characteristic function: 



^W = i-i/;Me-.M}t^ 



All of the needed statistical parameters can be found by using this equation, although this is a 
tricky integral for two reasons: the integral is an oscillatory integral, and the upper limit is infinity. 
These problems can be overcome by using Q-precision Gauss-Legendre quadrature and determining an 
"effective infinity" related to the number of computed spectra. The appropriate characteristic function, 
which is quite complicated, is discussed in some detail by Deans, Cullers, and Stauduhar (ref. 2). 



USE OF THE TABLES 

Use of the tables is straightforward. The quantity m represents the number of computed spectra. 
This is related to the number n of independent spectra by m = 2n - I when there is a 50% overlap 
and end effects are included. (Here it is assumed that the probability density function for the n sample 
average of the output of a square-law detector is a noncentral chi-square distribution with 2n degrees 
of freedom.) If r represents the threshold, then the false alarm probability is given by 

f{x)dx 

where f{x) is the central overlapped Hann probability density function with 2m degrees of freedom, 
normalized so the mean noise power is unity. In the tables this threshold is expressed in decibels (dB), 



r(dB) = 101ogioT 

Given the threshold, the detection probability can be computed in terms of the signal-to-noise ratio 
r. This ratio is often expressed in decibels, 

SNR(dB) = lOlogior 

The detection probability is found by integrating the noncentral probability density function for 
overlapped Hann spectra, 

roo 



Pd = J 9{x,r)dx 



EXAMPLE. Suppose Pfa = 10"^^ and m = 500. The signal-to-noise ratio that yields a detection 
probability of 0.50 is -4.4235 dB. 



REFERENCES 

1. Cullers, D. K.; Linscott, I. R.; and Oliver, B. M,; Signal Processing in SETI. Comm. of the ACM, 

vol.28, 1985, pp. 1151-1163. 

2. Deans, S. R.; Cullers, D. K.; and Stauduhar, R.: Computational Problems and Signal Processing 

in SETI. Maximum Entropy and Bayesian Methods, W. T. Grandy, Jr., and L. H. Schick, eds., 
Kluwer Academic Publishers, Dordrecht, 1991, pp. 67-75. 

3. Harris, F. J: On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform. 

Proc. of the ffiEE, vol. 66, 1978, pp. 51-83. 

4. Johnson, N. L.; and Kotz, S.: Continuous Univariate Distributions-2, Wiley, New York, 1970. 



-in-l 



Table 1. Overlap Hann, Pfa = 10 



Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pd = 0.50 


Pd = 0,70 


Pd = 0.90 


Pd = 0.99 


A 


2.2584 


-0.8806 


1.1946 


3.4293 


5.7244 


5 


2.0699 


-1.4406 


0.6005 


2.7973 


5.0544 


6 


1.9242 


-1.8966 


0.1166 


2.2825 


4.5083 


8 


1.7096 


-2.6117 


-0.6419 


1.4754 


3.6516 


10 


1.5561 


-3.1619 


-1.2250 


0.8552 


2.9932 


13 


1.3900 


-3.8034 


-1.9038 


0.1338 


2.2272 


16 


1.2693 


-4.3066 


-2.4355 


-0.4307 


1.6282 


20 


1.1495 


-4.8431 


-3.0013 


-1.0305 


0.9920 


25 


1.0397 


-5.3752 


-3.5612 


-1.6232 


0.3639 


32 


0.9292 


-5.9589 


-4.1740 


-2.2706 


-0.3212 


40 


0.8385 


-6.4823 


-4.7221 


-2.8484 


-0.9318 


50 


0.7559 


-7.0020 


-5.2650 


-3.4193 


-1.5342 


64 


0.6734 


-7.5728 


-5.8597 


-4.0432 


-2.1912 


80 


0.6061 


-8.0855 


-6.3923 


-4.6006 


-2.7770 


100 


0.5452 


-8.5953 


-6.9205 


-5.1520 


-3.3552 


125 


0.4901 


-9.1024 


-7.4447 


-5.6977 


-3.9263 


160 


0.4353 


-9.6607 


-8.0203 


-6.2954 


-4.5501 


200 


0.3909 


-10.1631 


-8.5370 


-6.8305 


-5.1073 


250 


0.3509 


-10.6635 


-9.0505 


-7.3610 


-5.6584 


320 


0.3112 


-11.2152 


-9.6153 


-7.9431 


-6.2616 


400 


0.2792 


-11.7122 


-10.1231 


-8.4652 


-6.8013 


500 


0.2503 


-12.2079 


-10.6286 


-8.9838 


-7.3361 


640 


0.2218 


-12.7549 


-11.1854 


-9.5537 


-7.9226 


800 


0.1988 


-13.2482 


-11.6867 


-10.0658 


-8.4483 


1000 


0.1781 


-13.7406 


-12.1863 


-10.5752 


-8.9702 


1250 


0.1596 


-14.2321 


-12.6844 


-11.0821 


-9.4887 


1600 


0.1413 


-14.7749 


-13.2337 


-11.6404 


-10.0585 


2000 


0.1265 


-15.2648 


-13.7289 


-12.1429 


-10.5704 



— in-2 



Table 2. Overlap Hann, Pfa = 10 



Spectra 


Threshold 


Signal- 


■to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pd = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


4.0868 


2.3229 


3.7076 


5.3575 


7.1906 


5 


3.7431 


1.7126 


3.0715 


4.6919 


6.4945 


6 


3.4781 


1.2180 


2.5554 


4.1513 


5.9284 


8 


3.0890 


0.4462 


1.7497 


3.3066 


5.0425 


10 


2.8114 


-0.1445 


1.1329 


2.6594 


4.3631 


13 


2.5113 


-0.8300 


0.4174 


1.9086 


3.5743 


16 


2.2932 


-1.3653 


-0.1411 


1.3226 


2.9585 


20 


2.0772 


-1.9338 


-0.7337 


0.7012 


2.3054 


25 


1.8791 


-2.4954 


-1.3185 


0.0884 


1.6616 


32 


1.6797 


-3.1090 


-1.9567 


-0.5796 


0.9602 


40 


1.5161 


-3.6571 


-2.5258 


-1.1746 


0.3359 


50 


1.3672 


-4.1994 


-3.0880 


-1.7615 


-0.2792 


64 


1.2183 


-4.7930 


-3.7024 


-2.4017 


-0.9494 


80 


1.0968 


-5.3243 


-4.2512 


-2.9727 


-1.5462 


100 


0.9868 


-5.8511 


-4.7944 


-3.5367 


-2.1347 


125 


0.8873 


-6.3736 


-5.3322 


-4.0941 


-2.7153 


160 


0.7883 


-6.9473 


-5.9214 


-4.7035 


-3,3490 


200 


0.7081 


-7.4622 


-6.4493 


-5.2484 


-3.9144 


250 


0.6357 


-7.9740 


-6.9731 


-5.7879 


-4.4731 


320 


0.5640 


-8.5369 


-7.5482 


-6.3790 


-5.0841 


400 


0.5059 


-9.0431 


-8.0644 


-6.9086 


-5.6303 


500 


0.4537 


-9.5470 


-8.5775 


-7.4340 


-6.1711 


640 


0.4021 


-10.1021 


-9.1418 


-8.0108 


-6.7636 


800 


0.3604 


-10.6020 


-9.6493 


-8.5285 


-7.2944 


1000 


0.3230 


-11.1003 


-10.1544 


-9.0430 


-7.8209 


1250 


0.2894 


-11.5971 


-10.6575 


-9.5546 


-8.3436 


1600 


0.2562 


-12.1452 


-11.2118 


-10.1175 


-8.9176 


2000 


0.2295 


-12.6395 


-11.7111 


-10.6239 


-9.4331 



Table 3. Overlap Hann, Pfa = 10 



-3 



Spectra 


Threshold 


Signal- 


■to-Noise Ratio (dB) for given P<j 


m 


(dB) 


Pd = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


5.2749 


4.0092 


5.1376 


6.5319 


8.1344 


5 


4.8425 


3.3618 


4.4698 


5.8404 


7.4177 


6 


4.5076 


2.8385 


3.9293 


5.2798 


6.8355 


8 


4.0137 


2.0244 


3.0875 


4.4054 


5.9259 


10 


3.6598 


1.4033 


2.4448 


3.7369 


5.2294 


13 


3.2758 


0.6850 


1.7012 


2.9629 


4.4221 


16 


2.9958 


0.1257 


1.1221 


2.3599 


3.7926 


20 


2.7176 


-0.4666 


0.5090 


1.7215 


3.1258 


25 


2.4619 


-1.0500 


-0.0948 


1.0929 


2.4692 


32 


2.2038 


-1.6857 


-0.7521 


0.4089 


1.7548 


40 


1.9914 


-2.2520 


-1.3371 


-0.1995 


1.1196 


50 


1.7977 


-2.8109 


-1.9139 


-0.7987 


0.4944 


64 


1.6036 


-3.4212 


-2.5429 


-1.4514 


-0.1861 


80 


1.4450 


-3.9661 


-3.1038 


-2.0328 


-0.7915 


100 


1.3011 


-4.5053 


-3.6580 


-2.6063 


-1.3880 


125 


1.1707 


-5.0391 


-4.2059 


-3.1724 


-1.9760 


160 


1.0409 


-5.6239 


-4.8053 


-3.7907 


-2.6172 


200 


0.9355 


-6.1479 


-5.3415 


-4.3428 


-3.1889 


250 


0.8403 


-6.6680 


-5.8728 


-4.8890 


-3.7533 


320 


0.7459 


-7.2391 


-6.4553 


-5.4869 


-4.3701 


400 


0.6695 


-7.7519 


-6.9777 


-6.0220 


-4.9212 


500 


0.6006 


-8.2618 


-7.4963 


-6.5524 


-5.4664 


640 


0.5325 


-8.8228 


-8.0662 


-7.1342 


-6.0634 


800 


0.4774 


-9.3275 


-8.5781 


-7.6560 


-6.5979 


1000 


0.4280 


-9.8300 


-9.0873 


-8.1743 


-7.1279 


1250 


0.3835 


-10.3307 


-9.5941 


-8.6894 


-7.6536 


1600 


0.3397 


-10.8826 


-10.1520 


-9.2557 


-8.2308 


2000 


0.3043 


-11.3799 


-10.6543 


-9.7648 


-8.7488 



Table 4. Overlap Hann, Pf^ = 10 



_ in-4 



Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pd = 0.50 


Pi = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


6.1742 


5.1775 


6.1591 


7.3974 


8.8499 


5 


5.6805 


4.5017 


5.4667 


6.6852 


8.1162 


6 


5.2967 


3.9560 


4.9068 


6.1083 


7.5208 


8 


4.7282 


3.1086 


4.0360 


5.2096 


6.5913 


10 


4.3191 


2.4634 


3.3723 


4.5235 


5.8804 


13 


3.8735 


1.7188 


2.6057 


3.7301 


5.0573 


16 


3.5476 


1.1402 


2.0098 


3.1129 


4.4161 


20 


3.2228 


0.5287 


1.3798 


2.4602 


3.7376 


25 


2.9235 


-0.0724 


0.7605 


1.8184 


3.0701 


32 


2.6205 


-0.7260 


0.0874 


1.1209 


2.3445 


40 


2.3707 


-1.3071 


-0.5107 


0.5013 


1.7000 


50 


2.1423 


-1.8795 


-1.0995 


-0.1082 


1.0662 


64 


1.9130 


-2.5034 


-1.7407 


-0.7715 


0.3769 


80 


1.7252 


-3.0594 


-2.3116 


-1.3615 


-0.2358 


100 


1.5546 


-3.6087 


-2.8749 


-1.9429 


-0.8391 


125 


1.3998 


-4.1516 


-3.4311 


-2.5164 


-1.4333 


160 


1.2455 


-4.7457 


-4.0388 


-3.1420 


-2.0808 


200 


1.1200 


-5.2771 


-4.5818 


-3.7002 


-2.6577 


250 


1.0066 


-5.8039 


-5.1193 


-4.2519 


-3.2270 


320 


0.8940 


-6.3817 


-5.7080 


-4.8554 


-3.8487 


400 


0.8027 


-6.8999 


-6.2354 


-5.3950 


-4.4038 


500 


0.7204 


-7.4147 


-6.7586 


-5.9296 


-4.9528 


640 


0.6389 


-7.9805 


-7.3331 


-6.5156 


-5.5535 


800 


0.5730 


-8.4891 


-7.8486 


-7.0409 


-6.0911 


1000 


0.5138 


-8.9952 


-8.3612 


-7.5622 


-6.6238 


1250 


0.4606 


-9.4990 


-8.8709 


-8.0801 


-7.1521 


1600 


0.4080 


-10.0541 


-9.4319 


-8.6492 


-7.7319 


2000 


0.3656 


-10.5539 


-9.9365 


-9.1606 


-8.2520 





Table 5. Overlap 


Hann, Pj^ = 


10-^ 




Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pd = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


6.9043 


6.0788 


6.9615 


8.0903 


9.4334 


5 


6.3646 


5.3802 


6.2489 


7.3609 


8.6855 


6 


5.9435 


4.8164 


5.6730 


6.7704 


8.0787 


8 


5.3175 


3.9416 


4.7781 


5.8511 


7.1321 


10 


4.8653 


3.2765 


4.0968 


5.1499 


6.4088 


13 


4.3711 


2.5099 


3.3108 


4.3400 


5.5718 


16 


4.0085 


1.9152 


2.7005 


3.7104 


4.9204 


20 


3.6463 


1.2875 


2.0561 


3.0453 


4.2315 


25 


3.3116 


0.6713 


1.4235 


2.3920 


3.5544 


32 


2.9721 


0.0024 


0.7367 


1.6828 


2.8190 


40 


2.6915 


-0.5912 


0.1273 


1.0534 


2.1662 


50 


2.4345 


-1.1752 


-0.4719 


0.4348 


1.5247 


64 


2.1760 


-1.8107 


-1.1235 


-0.2376 


0.8277 


80 


1.9639 


-2.3763 


-1.7031 


-0.8352 


0.2085 


100 


1.7710 


-2.9342 


-2.2743 


-1.4236 


-0.4007 


125 


1.5957 


-3.4851 


-2.8377 


-2.0034 


-1.0004 


160 


1.4207 


-4.0870 


-3.4526 


-2.6355 


-1.6535 


200 


1.2783 


-4.6249 


-4.0015 


-3.1990 


-2.2350 


250 


1.1494 


-5.1574 


-4.5443 


-3.7555 


-2.8086 


320 


1.0212 


-5.7410 


-5.1385 


-4.3638 


-3.4346 


400 


0.9173 


-6.2639 


-5.6702 


-4.9075 


-3.9932 


500 


0.8236 


-6.7829 


-6.1974 


-5.4457 


-4.5454 


640 


0.7307 


-7.3530 


-6.7757 


-6.0353 


-5.1494 


800 


0.6555 


-7.8649 


-7.2946 


-6.5635 


-5.6896 


1000 


0.5879 


-8.3740 


-7.8100 


-7.0876 


-6.2248 


1250 


0.5271 


-8.8806 


-8.3223 


-7.6079 


-6.7554 


1600 


0.4670 


-9.4383 


-8.8859 


-8.1794 


-7.3374 


2000 


0.4185 


-9.9403 


-9.3926 


-8.6928 


-7.8394 



— in-6 



Table 6. Overlap Hann, Pja = 10 



Spectra 


Threshold 


Signal- 


to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Fd = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


7.5217 


6.8158 


7.6255 


8.6715 


9.9294 


5 


6.9458 


6.0983 


6.8961 


7.9276 


9.1693 


6 


6.4949 


5.5194 


6.3067 


7.3254 


8.5527 


8 


5.8224 


4.6216 


5.3913 


6.3882 


7.5912 


10 


5.3349 


3.9394 


4.6947 


5.6739 


6.8568 


13 


4.8007 


3.1540 


3.8919 


4.8494 


6.0075 


16 


4.4077 


2.5453 


3.2690 


4.2090 


5.3470 


20 


4.0141 


1.9035 


2.6121 


3.5330 


4.6489 


25 


3.6497 


1.2742 


1.9676 


2.8694 


3.9631 


32 


3.2791 


0.5920 


1.2689 


2.1497 


3.2187 


40 


2.9723 


-0.0128 


0.6494 


1.5115 


2.5586 


50 


2.6908 


-0.6069 


0.0411 


0.8849 


1.9102 


64 


2.4071 


-1.2527 


-0.6199 


0.2042 


1.2060 


80 


2.1741 


-1.8267 


-1.2072 


-0.4001 


0.5810 


100 


1.9618 


-2.3923 


-1.7854 


-0.9948 


-0.0336 


125 


1.7687 


-2.9502 


-2.3552 


-1.5803 


-0.6383 


160 


1.5757 


-3.5591 


-2.9766 


-2.2182 


-1.2964 


200 


1.4184 


-4.1026 


-3.5308 


-2.7864 


-1.8821 


250 


1.2759 


-4.6404 


-4.0784 


-3.3473 


-2.4594 


320 


1.1342 


-5.2290 


-4.6773 


-3.9600 


-3.0893 


400 


1.0191 


-5.7561 


-5.2129 


-4.5072 


-3.6511 


500 


0.9153 


-6.2789 


-5.7436 


-5.0486 


-4.2062 


640 


0.8123 


-6.8527 


-6.3255 


-5.6415 


-4.8132 


800 


0.7289 


-7.3676 


-6.8472 


-6.1724 


-5.3558 


1000 


0.6539 


-7.8795 


-7.3652 


-6.6989 


-5.8933 


1250 


0.5864 


-8.3884 


-7.8798 


-7.2213 


-6.4258 


1600 


0.5197 


-8.9486 


-8.4457 


-7.7950 


-7.0099 


2000 


0.4658 


-9.4526 


-8.9543 


-8.3102 


-7.5335 





Table 7. Overlap 


Hann, Pfa = 


10-/ 




Spectra 


Threshold 


Signal- 


to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pa = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


8.0581 


7.4409 


8.1937 


9.1738 


10.3626 


5 


7.4525 


6.7075 


7.4500 


8.4174 


9.5919 


6 


6.9771 


6.1157 


6.8490 


7.8051 


8.9667 


8 


6.2659 


5.1979 


5.9156 


6.8523 


7.9919 


10 


5.7488 


4.5010 


5.2058 


6.1264 


7.2476 


13 


5.1805 


3.6991 


4.3880 


5.2888 


6.3873 


16 


4.7615 


3.0780 


3.7541 


4.6387 


5.7185 


20 


4.3410 


2.4238 


3.0858 


3.9528 


5.0120 


25 


3.9508 


1.7828 


2.4309 


3.2800 


4.3183 


32 


3.5533 


1.0887 


1.7213 


2.5507 


3.5658 


40 


3.2236 


0.4740 


1.0928 


1.9047 


2.8988 


50 


2.9206 


-0.1293 


0.4761 


1.2706 


2.2441 


64 


2.6147 


-0.7843 


-0.1933 


0.5825 


1.5335 


80 


2.3632 


-1.3660 


-0.7875 


-0.0281 


0.9031 


100 


2.1337 


-1.9385 


-1.3721 


-0.6284 


0.2836 


125 


1.9247 


-2.5027 


-1.9478 


-1.2192 


-0.3258 


160 


1.7156 


-3.1179 


-2.5750 


-1.8623 


-0.9885 


200 


1.5450 


-3.6666 


-3.1339 


-2.4349 


-1.5780 


250 


1.3904 


-4.2090 


-3.6859 


-2.9998 


-2.1589 


320 


1.2365 


-4.8024 


-4.2892 


-3.6164 


-2.7923 


400 


1.1114 


-5.3332 


-4.8284 


-4.1668 


-3.3570 


500 


0.9985 


-5.8594 


-5.3623 


-4.7113 


-3.9148 


640 


0.8864 


-6.4365 


-5.9474 


-5.3071 


-4.5244 


800 


0.7956 


-6.9543 


-6.4717 


-5.8404 


-5.0693 


1000 


0.7139 


-7.4686 


-6.9920 


-6.3691 


-5.6087 


1250 


0.6403 


-7.9798 


-7.5088 


-6.8936 


-6.1432 


1600 


0.5676 


-8.5421 


-8.0768 


-7.4693 


-6.7290 


2000 


0.5088 


-9.0478 


-8.5871 


-7.9860 


-7.2542 



10 



— in-8 



Table 8. Overlap Hann, Pfa = 10 



Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pi = 0.50 


Pi = 0.70 


Pi = 0.90 


Pi = 0.99 


4 


8.5331 


7.9844 


8.6911 


9.6172 


10.7482 


5 


7.9028 


7.2374 


7.9351 


8.8499 


9.9682 


6 


7.4067 


6.6344 


7.3240 


8.2286 


9.3354 


8 


6.6624 


5.6993 


6.3748 


7.2620 


8.3487 


10 


6.1198 


4.9893 


5.6532 


6.5257 


7.5954 


13 


5.5221 


4.1727 


4.8221 


5.6764 


6.7251 


16 


5.0804 


3.5406 


4.1782 


5.0175 


6.0487 


20 


4.6363 


2.8752 


3.4998 


4.3226 


5.3345 


25 


4.2234 


2.2237 


2.8352 


3.6413 


4.6336 


32 


3.8021 


1.5188 


2.1158 


2.9033 


3.8735 


40 


3.4520 


0.8951 


1.4791 


2.2500 


3.2002 


50 


3.1298 


0.2835 


0.8548 


1.6091 


2.5396 


64 


2.8041 


-0.3799 


0.1776 


0.9141 


1.8230 


80 


2.5358 


-0.9686 


-0.4230 


0.2978 


1.1877 


100 


2.2909 


-1.5475 


-1.0134 


-0.3078 


0.5635 


125 


2.0675 


-2.1175 


-1.5945 


-0.9034 


-0.0501 


160 


1.8439 


-2.7385 


-2.2271 


-1.5515 


-0.7171 


200 


1.6612 


-3.2920 


-2.7904 


-2.1280 


-1.3102 


250 


1.4956 


-3.8386 


-3.3465 


-2.6965 


-1.8943 


320 


1.3305 


-4.4363 


-3.9537 


-3.3168 


-2.5310 


400 


1.1963 


-4.9706 


-4.4962 


-3.8703 


-3.0984 


500 


1.0750 


-5.4999 


-5.0331 


-4.4175 


-3.6587 


640 


0.9546 


-6.0802 


-5.6211 


-5.0161 


-4.2709 


800 


0.8570 


-6.6004 


-6.1478 


-5.5516 


-4.8178 


1000 


0.7692 


-7.1170 


-6.6703 


-6.0823 


-5.3591 


1250 


0.6900 


-7.6303 


-7.1891 


-6.6086 


-5.8952 


1600 


0.6117 


-8.1946 


-7.7590 


-7.1862 


-6.4828 


2000 


0.5485 


-8.7020 


-8.2709 


-7.7045 


-7.0094 



11 





Table 9. Overlap 


Hann, Pj^ = 


10-^ 




Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pi = 0.50 


Pi = 0.70 


Pi = 0.90 


Pi = 0.99 


4 


8.9599 


8.4657 


9.1342 


10.0146 


11.0964 


5 


8.3084 


7.7070 


8.3674 


9.2378 


10.3080 


6 


7.7945 


7.0941 


7.7473 


8.6086 


9.6684 


8 


7.0217 


6.1436 


6.7842 


7.6296 


8.6710 


10 


6.4568 


5.4220 


6.0519 


6.8838 


7.9096 


13 


5.8333 


4.5922 


5.2088 


6.0239 


7.0300 


16 


5.3714 


3.9502 


4.5558 


5.3569 


6.3467 


20 


4.9063 


3.2746 


3.8681 


4.6538 


5.6254 


25 


4.4732 


2.6136 


3.1948 


3.9647 


4.9177 


32 


4.0304 


1.8988 


2.4663 


3.2187 


4.1507 


40 


3.6620 


1.2668 


1.8220 


2.5585 


3.4715 


50 


3.3224 


0.6475 


1.1907 


1.9114 


2.8054 


64 


2.9787 


-0.0237 


0.5063 


1.2099 


2.0832 


80 


2.6953 


-0.6187 


-0.1002 


0.5883 


1.4431 


100 


2.4362 


-1.2035 


-0.6961 


-0.0222 


0.8146 


125 


2.1997 


-1.7789 


-1.2821 


-0.6224 


0.1971 


160 


1.9627 


-2.4053 


-1.9197 


-1.2749 


-0.4740 


200 


1.7690 


-2.9632 


-2.4872 


-1.8553 


-1.0704 


250 


1.5931 


-3.5138 


-3.0470 


-2.4272 


-1.6576 


320 


1.4178 


-4.1154 


-3.6580 


-3.0509 


-2.2974 


400 


1.2751 


-4.6530 


-4.2035 


-3.6072 


-2.8673 


500 


1.1462 


-5.1853 


-4.7432 


-4.1570 


-3.4299 


640 


1.0181 


-5.7684 


-5.3340 


-4.7582 


-4.0444 


800 


0.9142 


-6.2910 


-5.8629 


-5.2958 


-4.5933 


1000 


0.8206 


-6.8098 


-6.3875 


-5.8285 


-5.136 


1250 


0.7364 


-7.3249 


-6.9080 


-6.3564 


-5.6741 


1600 


0.6529 


-7.8912 


-7.4798 


-6.9357 


-6.2632 


2000 


0.5855 


-8.4001 


-7.9931 


-7.4554 


-6.7911 



12 



Table 10. Overlap Hann, P/a = ^0 



-10 



Spectra 


Threshold 


Signal- 


to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pd = 0.50 


Pi = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


9.3475 


8.8980 


9.5339 


10.3752 


11.4140 


5 


8.6779 


8.1289 


8.7576 


9.5899 


10.6184 


6 


8.1485 


7.5074 


8.1296 


8.9536 


9.9725 


8 


7.3506 


6.5431 


7.1538 


7.9633 


8.9654 


10 


6.7661 


5.8111 


6.4120 


7.2090 


8.1966 


13 


6.1194 


4.9693 


5,5579 


6.3393 


7.3085 


16 


5.6396 


4.3182 


4.8966 


5.6649 


6.6187 


20 


5.1556 


3.6333 


4.2003 


4.9542 


5.8908 


25 


4.7042 


2.9635 


3.5190 


4.2579 


5.1768 


32 


4.2420 


2.2396 


2.7821 


3.5044 


4.4033 


40 


3.8568 


1.6000 


2.1308 


2.8379 


3.7185 


50 


3.5014 


0.9736 


1.4929 


2.1849 


3.0473 


64 


3.1412 


0.2952 


0.8019 


1.4774 


2.3198 


80 


2.8438 


-0.3059 


0.1898 


0.8508 


1.6754 


100 


2.5717 


-0.8961 


-0.4112 


0.2356 


1.0428 


125 


2.3231 


-1.4765 


-1.0019 


-0.3688 


0.4215 


160 


2.0738 


-2.1079 


-1.6442 


-1.0256 


-0.2534 


200 


1.8697 


-2.6700 


-2.2156 


-1.6095 


-0.8530 


250 


1.6845 


-3.2244 


-2.7788 


-2.1846 


-1.4430 


320 


1.4996 


-3.8297 


-3.3934 


-2.8116 


-2.0857 


400 


1.3491 


-4.3704 


-3.9418 


-3.3706 


-2.6581 


500 


1.2129 


-4.9053 


-4.4840 


-3.9227 


-3.2229 


640 


1.0777 


-5.4912 


-5.0774 


-4.5263 


-3.8396 


800 


0.9679 


-6.0159 


-5.6083 


-5.0658 


-4.3902 


1000 


0.8691 


-6.5363 


-6.1345 


-5.6000 


-4.9347 


1250 


0.7799 


-7.0537 


-6.6572 


-6.1300 


-5.4742 


1600 


0.6916 


-7.6217 


-7.2306 


-6.7109 


-6.0649 


2000 


0.6202 


-8.1320 


-7.7454 


-7.2318 


-6.5940 



13 





Table 11. Overlap] 


Hfann, Pfa = 


10-^' 




Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pi = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


9.7028 


9.2905 


9.8983 


10.7054 


11.7065 


5 


9.0173 


8.5123 


9.1136 


9.9126 


10.9042 


6 


8.4743 


7.8830 


8.4784 


9.2699 


10.2528 


8 


7.6541 


6.9064 


7.4913 


8.2694 


9.2367 


10 


7.0520 


6.1648 


6.7406 


7.5073 


8.4611 


13 


6.3847 


5.3122 


5.8766 


6.6286 


7.5652 


16 


5.8887 


4.6527 


5.2076 


5.9473 


6.8694 


20 


5.3876 


3.9593 


4.5034 


5.2295 


6.1352 


25 


4.9195 


3.2813 


3.8146 


4.5265 


5.4154 


32 


4.4395 


2.5489 


3.0699 


3.7659 


4.6357 


40 


4.0390 


1.9022 


2.4120 


3.0935 


3.9458 


50 


3.6689 


1.2692 


1.7680 


2.4350 


3.2696 


64 


3.2935 


0.5840 


1.0707 


1.7218 


2.5372 


80 


2.9832 


-0.0226 


0.4534 


1.0904 


1.8885 


100 


2.6989 


-0.6181 


-0.1524 


0.4709 


1.2521 


125 


2.4391 


-1.2031 


-0.7475 


-0.1375 


0.6272 


160 


2.1782 


-1.8393 


-1.3943 


-0.7984 


-0.0513 


200 


1.9646 


-2.4052 


-1.9693 


-1.3856 


-0.6538 


250 


1.7705 


-2.9632 


-2.5358 


-1.9638 


-1.2467 


320 


1.5766 


-3.5721 


-3.1538 


-2.5939 


-1.8922 


400 


1.4188 


-4.1155 


-3.7048 


-3.1553 


-2.4668 


500 


1.2759 


-4.6531 


-4.2495 


-3.7097 


-3.0337 


640 


1.1339 


-5.2416 


-4.8453 


-4.3156 


-3.6525 


800 


1.0186 


-5.7685 


-5.3784 


-4.8571 


-4.2048 


1000 


0.9147 


-6.2911 


-5.9066 


-5.3932 


-4.7511 


1250 


0.8210 


-6.8097 


-6.4306 


-5.9244 


-5.2918 


1600 


0.7282 


-7.3794 


-7.0056 


-6.5068 


-5.8839 


2000 


0.6531 


-7.8911 


-7.5217 


-7.0290 


-6.4142 



14 



Table 12. Overlap Hann, P/a = 10 



- in-l2 



Spectra 


Threshold 


Signal-to-Noise Ratio (dB) for given Pd 


m 


(dB) 


Pi = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


10.0309 


9.6501 


10.2331 


11.0101 


11.9776 


5 


9.3314 


8.8637 


9.4410 


10.2105 


11.1694 


6 


8.7763 


8.2275 


8.7993 


9.5621 


10.5129 


8 


7.9362 


7.2397 


7.8019 


8.5523 


9.4886 


10 


7.3183 


6.4894 


7.0432 


7.7829 


8.7067 


13 


6.6323 


5.6268 


6.1699 


6.8960 


7.8035 


16 


6.1214 


4.9596 


5.4938 


6.2083 


7.1021 


20 


5.6046 


4.2582 


4.7823 


5.4838 


6.3621 


25 


5.1213 


3.5727 


4.0865 


4.7745 


5.6367 


32 


4.6249 


2.8324 


3.3345 


4.0074 


4.8512 


40 


4.2102 


2.1790 


2.6704 


3.3294 


4.1564 


50 


3.8266 


1.5399 


2.0206 


2.6656 


3.4756 


64 


3.4370 


0.8483 


1.3174 


1.9471 


2.7384 


80 


3.1146 


0.2363 


0.6952 


1.3112 


2.0858 


100 


2.8191 


-0.3640 


0.0848 


0.6876 


1.4457 


125 


2.5487 


-0.9535 


-0.5144 


0.0754 


0.8174 


160 


2.2770 


-1.5942 


-1.1654 


-0.5894 


0.1354 


200 


2.0544 


-2.1637 


-1.7438 


-1.1798 


-0.4699 


250 


1.8520 


-2.7250 


-2.3136 


-1.7609 


-1.0654 


320 


1.6497 


-3.3372 


-2.9346 


-2.3938 


-1.7135 


400 


1.4849 


-3.8835 


-3.4883 


-2.9577 


-2.2903 


500 


1.3357 


-4.4235 


-4.0353 


-3.5143 


-2.8592 


640 


1.1873 


-5.0144 


-4.6335 


-4.1224 


-3.4800 


800 


1.0668 


-5.5433 


-5.1684 


-4.6656 


-4.0340 


1000 


0.9580 


-6.0677 


-5.6984 


-5.2033 


-4.5818 


1250 


0.8601 


-6.5880 


-6.2238 


-5.7360 


-5.1238 


1600 


0.7629 


-7.1593 


-6.8004 


-6.3199 


-5.7173 


2000 


0.6844 


-7.6722 


-7.3177 


-6.8433 


-6.2487 



15 



Table 13. Overlap Hann, Py^ = 10 



_ in-13 



Spectra 


Threshold 


Signal- 


to-Noise Rat 


io (dB) for given Pd 


m 


(dB) 


Pd = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


10.3357 


9.9818 


10.5431 


11.2932 


12.2305 


5 


9.6238 


9.1882 


9.7442 


10.4875 


11.4169 


6 


9.0578 


8.5457 


9.0967 


9.8338 


10.7557 


8 


8.1999 


7.5477 


8.0898 


8.8154 


9.7239 


10 


7.5676 


6.7895 


7.3238 


8.0394 


8.9361 


13 


6.8644 


5.9176 


6.4419 


7.1447 


8.0261 


16 


6.3401 


5.2434 


5.7592 


6.4511 


7.3194 


20 


5.8089 


4.5345 


5.0409 


5.7204 


6.5740 


25 


5.3114 


3.8420 


4.3385 


5.0052 


5.8433 


32 


4.7999 


3.0943 


3.5796 


4.2318 


5.0523 


40 


4.3719 


2.4346 


2.9097 


3.5485 


4.3528 


50 


3.9759 


1.7899 


2.2547 


2.8801 


3.6679 


64 


3.5728 


1.0920 


1.5456 


2.1562 


2.9259 


80 


3.2392 


0.4750 


0.9187 


1.5161 


2.2695 


100 


2.9331 


-0.1299 


0.3041 


0.8885 


1.6259 


125 


2.6528 


-0.7237 


-0.2991 


0.2727 


0.9944 


160 


2.3709 


-1.3686 


-0.9541 


-0.3958 


0.3091 


200 


2.1398 


-1.9416 


-1.5358 


-0.9892 


-0.2990 


250 


1.9295 


-2.5061 


-2.1086 


-1.5731 


-0.8969 


320 


1.7193 


-3.1215 


-2.7326 


-2.2088 


-1.5476 


400 


1.5479 


-3.6703 


-3.2887 


-2.7750 


-2.1265 


500 


1.3927 


-4.2127 


-3.8380 


-3.3337 


-2.6973 


640 


1.2382 


-4.8060 


-4.4384 


-3.9438 


-3.3200 


800 


1.1127 


-5.3367 


-4.9752 


-4.4888 


-3.8756 


1000 


0.9994 


-5.8628 


-5.5068 


-5.0280 


-4.4248 


1250 


0.8973 


-6.3846 


-6.0337 


-5.5621 


-4.9681 


1600 


0.7961 


-6.9575 


-6.6118 


-6.1474 


-5.5629 


2000 


0.7143 


-7.4717 


-7.1303 


-6.6719 


-6.0954 



16 





Table 14. Overlap 


Hann, Pj^ = 


10-14 




Spectra 


Threshold 


Signal- 


to-Noise Ratio (dB) for given Pd | 


m 


(dB) 


Pd = 0.50 


Pd = 0.70 


Pd = 0.90 


Pd = 0.99 


4 


10.6204 


10.2899 


10.8316 


11.5575 


12.4675 


5 


9.8974 


9.4898 


10.0267 


10.7464 


11.6490 


6 


9.3217 


8.8416 


9.3739 


10.0878 


10.9836 


8 


8.4475 


7.8342 


8.3583 


9.0616 


9.9448 


10 


7.8021 


7.0686 


7.5854 


8.2794 


9.1515 


13 


7.0832 


6.1882 


6.6956 


7.3775 


8.2351 


16 


6.5464 


5.5073 


6.0068 


6.6783 


7.5235 


20 


6.0019 


4.7916 


5.2820 


5.9418 


6.7729 


25 


5.4913 


4.0924 


4.5734 


5.2210 


6.0373 


32 


4.9656 


3.3377 


3.8081 


4.4417 


5.2411 


40 


4.5254 


2.6721 


3.1327 


3.7534 


4.5371 


50 


4.1193 


2.0247 


2.4752 


3.0827 


3.8502 


64 


3.7020 


1.3182 


1.7581 


2.3515 


3.1017 


80 


3.3578 


0.6965 


1.1267 


1.7073 


2.4417 


100 


3.0416 


0.0872 


0.5080 


1.0759 


1.7947 


125 


2.7520 


-0.5105 


-0.0990 


0.4566 


1.1600 


160 


2,4604 


-1.1595 


-0.7578 


-0.2153 


0.4716 


200 


2.2213 


-1.7359 


-1.3427 


-0.8116 


-0.1391 


250 


2.0035 


-2.3036 


-1.9184 


-1.3983 


-0.7396 


320 


1.7858 


-2.9219 


-2.5452 


-2.0366 


-1.3926 


400 


1.6081 


-3.4732 


-3.1037 


-2.6049 


-1.9735 


500 


1.4471 


-4.0178 


-3.6551 


-3.1656 


-2.5461 


640 


1.2868 


-4.6133 


-4.2577 


-3.7778 


-3.1707 


800 


1.1566 


-5.1459 


-4.7961 


-4.3244 


-3.7278 


1000 


1.0391 


-5.6736 


-5.3293 


-4.8651 


-4.2784 


1250 


0.9331 


-6.1969 


-5.8577 


-5.4005 


-4.8229 


1600 


0.8279 


-6.7712 


-6.4372 


-5.9872 


-5.4190 


2000 


0.7429 


-7.2866 


-6.9568 


-6.5128 


-5.9524 



17 



REPORT DOCUMENTATION PAGE 



Form Approved 
OMB No. 0704-0188 



Publte reporting burden lor this collection of information Is ..timated to average 1 hour per response, inclydinfl the time for reviewing insJructjons^.warchjngeKisti^^^^ 'i^lTr.^^: 

■° -■--• —■' '-"ig and reviewing the collection of informatioi 

ifl this burden, to Washington Headquarters -.- .- ■-,„-_ ■■,,. ^ t„„.«n nr 'in<^^i'^ 

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1. AGENCY USE ONLY fUavr» 6/«n*; 



2. REPORT DATE 



September 1991 



3. REPORT TYPE AND DATES COVERED 

Technical Memorandum 



4. TITLE AND SUBTITLE 



Tables of Square-Law Signal Detection Statistics for Hann Spectra 
with 50% Overlap 



6. AUTHOR(S) 

Stanley R. Deans (University of South Florida) and D. Kent Cullers 



7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Ames Research Center 
Moffett Field, CA 94035-1000 



9. SPONSORING/MONITORING AGENCY NAME(S) AND AODRESS(ES) 

National Aeronautics and Space Administration 
Washington, DC 20546-0001 



5. FUNDING NUMBERS 



RTOP 108-41-60 



PERFORMING ORGANIZATION 
REPORT NUMBER 



A-91037 



10. SPONSORING/MONITORING 
AGENCY REPORT NUMBER 



NASA TM- 103830 



11. SUPPLEMENTARY NOTES 

Point of Contact: D. Kent Cullers, Ames Research Center, MS 239-22, MoflFett Field, CA 94035-1000 
(415) 604-6019 orFTS 464-6019 



12a. DISTRIBUTION/AVAILABILITY STATEMENT 

Unclassified-Unlimited 
Subject Category - 65 



I2b. DISTRIBUTION CODE 



13. ABSTRACT (Uaximum 200 words) 

The Search for Extraterrestrial Intelligence, currcntiy being planned by NASA, will require that an 
enormous amount of data be analyzed in real time by special-purpose hardware, and it is expected that 
overlapped Hann data windows will play an important role in this analysis. In order to understand the 
statistical implications of this approach, it has been necessary to compute detection statistics for overlapped 
Hann spectra. Tables of signal detection statistics are given for false alarm rates from lO"^"* to 10"^ and signal 
detection probabilities from 0.50 to 0.99; the number of computed spectra ranges fixjm 4 to 2000. 



14. SUBJECT TERMS 

Distribution of quadratic forms. Signal detection statistics 



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OF REPORT 



Unclassified 



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Unclassified 



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18 



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A02 



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